Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Vol. 10, Issue 2 (December 2015), pp. 878-892
Applications and Applied
Mathematics:
An International Journal
(AAM)
The shifted Jacobi polynomial integral operational matrix
for solving Riccati differential equation of fractional order
A. Neamaty1, B. Agheli2 and R. Darzi3
1
Department of Mathematics
University of Mazandaran
Babolsar, Iran
namaty@umz.ac.ir
2
Department of Mathematics
Qaemshahr Branch
Islamic Azad University
Qaemshahr, Iran
b.agheli@qaemshahriau.ac.ir
3
Department of Mathematics
Neka Branch, Islamic Azad University
Neka, Iran
r.darzi@iauneka.ac.ir
Received: October 31, 2014; Accepted: June 26, 2015
Abstract
In this article, we have applied Jacobi polynomial to solve Riccati differential equation of
fractional order. To do so, we have presented a general formula for the Jacobi operational matrix
of fractional integral operator. Using the Tau method, the solution of this problem reduces to the
solution of a system of algebraic equations. The numerical results for the examples presented in
this paper demonstrate the efficiency of the present method.
Keywords: Fractional differential equations; Operational matrix; Jacobi polynomials;
Tau method, Riccati equation
MSC 2010 No.: 34A08; 26A33; 47B36
1. Introduction
Fractional calculus has been one of the most fascinating issues that have attracted the attention of
large group of scholars, particularly in the fields of mathematics and engineering. This is due to
the fact that boundary value problems of fractional differential equation can be employed to
878
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
879
explain various natural phenomena. Many scholars and authors in different fields such as physics,
fluid flows, electrical networks, and viscoelasticity have attempted to introduce a model for these
phenomena through using fractional differential equation [Oldham and Spanier (1974), Ross
(1975), Kilbas et al. (2006), Podlubny (1999), Lakshmikantham et al. (2009)]. Interested readers
can check other books and papers in the related literature to get further information about
fractional calculus [Kilbas et al. (2006), Podlubny (1999)].
We know that most fractional differential equations do not lend themselves to accurate analytical
solutions. Consequently, we should use approximate and numerical techniques to find solutions
for fractional differential equations. Various methods have been employed in the last few decades
to find such solutions.
These methods include fractional partial differential equations and fractional integro-differential
equations containing fractional derivatives as Adomian decomposition method [Momani and
Noor (2006), Ray et al. (2006), Wang (2006)], Variational iteration method [Inc (2008), Odibat
and Momani (2006), Abbasbandy(2007)], Homotopy analysis method [Hashim (2009),
Zurigat(2010) ] and other methods [Kazemi (2011), Sweilam et al. (2012), Erjaee et al. (2011), ].
Attempts to find accurate and efficient methods to solve fractional Riccati equations have invited
a lot of active research projects. Scholars and authors have presented various analytical and
numerical methods for solving this equation. Analytical method includes the ADM and VIM,
Abbasbandy (2007). Another approach through which we can solve fractional Riccati equation is
to use HPM, Abbasbandy (2007).
In the present research, we have employed Jacobi orthogonal polynomials to find solutions to the
Riccati differential equation of fractional order
D  y( x)  a( x) y( x)  b( x) y 2 ( x)  g ( x) ,
y(0)  d ,
(1.1)
(1.2)
in which D  signifies caputo fractional derivative operator of order a(x) and b(x) and g (x)
stand for real functions on R . The purpose of this study is to generalize Jacobi integral
operational matrix to fractional calculus.
Thus, these matrices have been used along with the Tau method to reduce the solution of this
problem to the solution of a system of algebraic equation.
2. Preliminaries
In this section, several definitions of fractional calculus are presented. The definitions include the
Jacobi polynomials, the shifted Jacobi polynomials and some of their properties.
2.1. Fractional Calculus
Definition 1.
A real function f (x) , x  0 is considered to be in the space C , (  R) if there exists a real
number n(  ) ,so that f ( x)  xn f1 ( x) , where f1 ( x)  C0,  and it is said to be in the space Ck
if and only if f ( k )  C , k  N .
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A. Neamaty et al.
Definition 2.
The Riemann-Liouville fractional integral operator of order   0 , of a function f  C ,  1
is given by
1 x
( x  r ) 1 f (r ) dr ,

a
( )

I f ( x)  I 0 f ( x), I 0 f ( x)  f ( x) .
I a f ( x) 
Definition 3. The Caputo's fractional derivative of f is defined as
D f ( x)  I k  D k f ( x) 
x
1
( x  r ) k  1 f ( k ) (r ) dr , x  0,

( k   ) 0
where, f  Ck1 , k  1    k and k  N .
Property 1.
For k  1    k , k  N , f  Ck ,   1 and x  0 the following properties satisfy
i) Da I a f ( x)  f ( x) ,
k 1
ii ) I a Da f ( x)  f ( x)   f ( j ) (a  )
j 0
( x  a) j
.
j!
2.2. Jacobi polynomials
The Jacobi polynomials which are represented by J n , ( z ), are orthogonal with regard to the
weight function w( z)  (1  z) (1  z)  on the interval I   1,1 , i.e.,

1
1
J n , ( z ) J m , ( z ) dz   n ,  m,n ,
where
 n ,  
2   1 (n    1) (n    1)
n! (2n      1) (n      1)
(2.2)
and  m, n is the Kronecker function.
One can easily notice that the weight function w(z ) belongs to L1 ( I ) if and only if  ,   1.
The following three term-recurrence to relation results in the Jacobi polynomials
J 0 ,   1, J1 ,  
1
    2z  1    ,
2
2
(2.1)
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881
J n,1 ( z)  an z  bn  J n ,  ( z)  cn J n,1 ( z), n  1,
where
2n      12n      2 ,
2(n  1)n      1
2n      12n      2 ,
an 
2(n  1)n      1
2n      1n   n    .
cn 
(n  1)n      12n     
an 
The Jacobi polynomials J n ,  ( z ), of degree n are generated by
J
 ,
n
( z)  2
n
 n    n   
i
n i

z  1 z  1 .
i  n  i 
i 0 
n
 
(2.3)
2.3. The shifted Jacobi polynomials
As a result of changing variable z  2 x  1, we obtain new orthogonal polynomials Pn ,  ( x) with
weight function ws ,   ( x)  1  x  x  on the interval [0,1] which is called shifted Jacobi
polynomials. These polynomials have the following orthogonality properties
1
P
0
where n ,  
 n , 
2   1
 ,
n
( x) Pm ,  ( x)ws ,   ( x) dx  n ,   mn ,
(2.4)
.
From (2.3), we can write Pn ,  ( x) as follows:
 ,
n
P
 n    n   
i

x  1 x n i ,
( x)   
i  n  i 
i 0 
n
n i
 n    n    i 
j

  1 x ni .
Pn , ( x)   
i  n  i  j 
i 0 j 0 
(2.5)
(2.6)
From relations (2.5) and (2.6), we can easily notice that the following properties are satisfied.
Property 2.
nn   
.
Pn ,  (0)   1 
 n 
Property 3.
d i  ,
n      i  1   i ,   i
Pn ( x) 
Pn i
( x).
i
dx
n      1
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A. Neamaty et al.
Property 4. The shifted Jacobi polynomial can be achieved in the following form:
n
Pn ( x)   pi n  xi ,
 ,
i 0
in which
n i  n      1 n   
pi n   1 

 , i  0, 1,..., n.
i

 n  i 
  0,
Property 5. For

1
0
j
x  Pm ,  ( x) ws ,   ( x) dx   pl j B  l    1,  1,
(2.7)
l 0
where is B(t , s) Beta function.
2.4. The approximation of functions in the Sobolov space
Suppose   0,1, then for any r  N ( N is the set of all non-negative integers), the weighted
Sobolev space H r    () can be defined in the usual way, which indicates its inner product, semiws
,
norm and norm by
u, vw
 , 
s
Particularly,
,  r ,w ,   and 
s
r , ws ,  
L2ws   H 0ws ,  (), and 
, respectively.
ws , 


H wr  ,      f | f can be measured, 
s
f
r
2
r , ws , 
f
   kx f
2
r , ws , 
2
r , ws k , k 
k 0
  rx f
2
ws r , r 
r , ws , 
r , ws
 , 
.

 ,
,
.
r
Now we can suppose the function f  H w ,   in
s
P m, ,  ( x)  span  p0 ,  ( x), p1 ,  ( x),..., pm , 1 ( x) ,
as presented in the following formula:

f ( x)   ki pi ,   ( x).
i 0
in which the coefficients k i are generated by:
(2.7)
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
ki 
1

 ,
j
1
0
883
pi ,  ( x) f ( x) ws ,  dx, i  0,1,... .
(2.8)
In practice, only the first m- terms shifted Jacobi polynomials are taken into account. Then we
have:
m1
f ( x)   ki pi ,  ( x)  k T p,
(2.9)
i 0
with
K  k0 , k1,..., km 1  ,
T

(2.10)

P  p0 ,  ( x), p1 ,  ( x),..., pm , 1 ( x) .
(2.11)
In as much as P m,  ,  is a finite dimensional vector space, f has a unique best approximation
from P m,  ,  , say f m ( x)  Pm,  ,  that is:
 y  P m ,  ,  , f ( x)  f m ( x) w  f ( x)  y
s
ws
.
r
Guo and Wang (2004), came to the conclusion that for any f  H w ,   , r  N and 0    r , a
s
generic positive constant C independent of any function, m ,  and  exists so that:
f ( x)  f m ( x)
 , ws( , )
 C m  1m      1
 r
2
f ( x) r , w .
s
3. The operational matrix of fractional integral
We can express Riemann-Liouville fractional integral operator of order  of the vector p by:
I  P  Q(  ) P.
(3.1)
where Q (  ) is the m n operational matrix of Riemann-Liouville fractional integral of order  .
Theorem 3.1.
If Q (  ) is the m n operational matrix of Riemann-Liouville fractional integral of order  , then
the elements of this matrix are taken as:
 
Q    qi(,j )
m1
i , j 0
i
j
  pk(i ) pl( j )
k  0 l 0
(k  1) B(k  l      1,   1)
.
 j , (k    1)
Now, we define the error vector E , as

E  I P Q
( )
P.
The maximum norm of vector E is defined as follows (Guo and Wang (2004))
(3.2)
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A. Neamaty et al.
E

m

x 0
 L  m   
  

 B(  1,   1) ,   0
 m!   m  1  x0   m 

m
x 0
L

  B(  1,   1) ,   0,



m
!



m

1
 x0 

(3.3)
where x0  0 and L  max1  x0 , x0 .
4. Main Results
Lemma 4.1.
Let


T
K  k0 , k1,..., km 1  , P  p0 ,  ( x), p1 ,  ( x),..., pm , 1 ( x) .
Now if we suppose that Q (  ) is the same in Theorem (3.1). Then,
P PT (Q(  ) )T K  HP,
 
where H  hi , j
hi , j 
i , j  0,1,...,n 1
2   1
 j , 
(4.1)
, with
 n 1  n 1  , 
 
 ,
 ,
    pt ( x) qlt(  ) kl ws ,  dx,
p
(
x
)
p
(
x
)
j
0 i
 
 l 0  t 0
1
i, j  0,1,..., n  1 .
Proof:
We denote Pi ,  ( x)  pi , i  0,1,..., n  1. We have
 p0 
 p 
T
( ) T
P P (Q ) K   1  p0
  


 pn 1 
p0
( )
( )
 q00
q10
 qn( 1) 0   k0 
 ( )

( )
q01
q11
 qn( 1)1   k0 

 p0 
 

   
 ( )
 
( )
( )
q0 n 1 q1 n 1  qn 1 n 1  kn 1 
n 1
n 1
n 1


 
 
k
p
p
q

k
p
p
q



k
p0 pi qn 1 i 



0
0
i
0
i
1
0
i
1
i
n

1

i 0
i 0
i 0


n 1
n 1
n 1
 
 
 
 k

p
p
q

k
p
p
q



k
p
p
q
0  1 i 0i
1  1 i 1i
n 1  1 i n 1i

.
i 0
i 0
i 0



n 1
n 1
 n 1

k0  pn 1 pi q0 i   k1  pn 1 pi q1i     kn 1  pn 1 pi qn1 i 
 i  0

i 0
i 0
If we consider f   f0 , f1,, fn 1  with
AAM: Intern. J., Vol. 10, Issue 2 (December 2015)
885
n 1
n 1
n 1
i 0
i 0
i 0
f j  k0  pk pi q0i   k1  pk pi q1i     k j  pk pi qn1 i , j  0,1,..., n  1,
and applying (2.8)-(2.9), the following is obtained:
 n 1

  h0 j p j 
 nj 01


h1 j p j 
  HP,
P PT (Q (  ) )T K   
j 0


 n 1 

 h p 
 j  0 n 1 j j 
where
hi , j 
2   1
 j , 
 n 1  n 1  , 
 
 ,
 ,
    pt ( x) qlt(  ) kl ws ,  dx,
p
(
x
)
p
(
x
)
j
0 i
 
 l 0  t 0
1
i, j  0,1,..., n  1 .
The proof is complete.
Now, we consider the Riccati equation with fractional orders of the form
D y( x)  a y( x)  b y 2 ( x)  g ( x), 0    1,
y(0)  d ,
(4.2)
(4.3)
where a, b, d are real constant coefficients and D  stand for the Caputo fractional derivative of
order  .
Using Definition (3), we can rewrite Equation (4.2):
I 1 Dy( x)  a y( x)  b y 2 ( x)  g ( x), 0    1.
(4.4)
To solve problems (4.2)-(4.3) we approximate D y(x) and g (x) by the shifted Jacobi
polynomials as:
m 1
D y ( x)   ki pi ,  ( x)  K T P,
(4.5)
i 0
m 1
g ( x)   gi pi ,  ( x)  GT P.
(4.6)
i 0
From (4.5), we get



D y( x) dx  I K T P  K T IP 
x
0
(4.7)
Consequently,
y( x)  K T Q(1) P  y(0),
(4.8)
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and
y 2 ( x)  K T Q(1) P PT (Q(1) )T K  2K T Q(1) P y(0)   y(0) ,
(4.9)
D  y( x)  I 1 D y( x)  I 1 K T P  K T Q (1 ) P.
(4.10)
2
Moreover, we have
Using (4.6) and (4.8)-(4.10), problems (4.2)-(4.3) can be rewritten as:

 

K T Q(1  ) P  a K T Q(1) P  y(0)  b K T Q(1) P PT (Q(1) )T K  2K T Q(1) Py (0)   y(0)  GT P.
2
Applying lemma (4.1), this relation reduces to the following relation


K T Q (1 ) P  a K T Q (1) P  C1  b  K T Q (1) HP 
(4.11)
 2 K Q C2 P  C3 P  G P,
T
(1)
T
where y(0)  C1P , Py (0)  C2 P and  y(0)2  C3 P can be calculated in the same way as (4.1).
By applying the typical Tau method see Canuto et al. (1988), a system of algebraic equation
KT L  F,
(4.12)
L  Q(1  )  a Q(1)  b Q(1) H  Q(1) C2 P,
(4.13)
F  GT  C1  C3 ,
(4.14)
is obtained.
5. Numerical results
In this section, we applied the method presented in this paper and solved some examples. The
examples reported in this section were selected from a large collection of problem to which this
method could be applied.
Example 1.
We consider the following fractional Riccati differential equation
D y( x)  2 y( x)  y 2 ( x)  g ( x),
with initial condition
0  x  1, 0    1
y(0)  0.
(5.1)
(5.2)
The exact solution of this problem for   1 was found to be of the form

1  2  1  
 .
y ( x)  1  2 tanh 2 x  log


2
2

1



By using the method that was elaborated in previous section, we have the approximations (4.5)(4.10).
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887
Using (4.11) and Tau method, the problems (5.1)-(5.2) are transformed to the following relation


K T Q(1  )  2Q(1)  Q(1) H  GT .
Letting m  2 ,     1and  
(5.3)
1
, we obtain
2
 
 
0.773748 0.21493 
 0.5 0.25
1
Q 2   
, Q1  
, G   ,


0.401202 0.351704
 0.4 0.0 
0
1
0.773748k0  0.687780k1 0.967185k0  0.859725k1 
H 
.
0.773748k0  0.687780k1 0.96718k0  0.859725k1 
Now, from (5.3) we conclude that K  6.91457
8.40791 . So
T
y( x)  1.0000 x  6.1391 107.
Figure1. The approximate solution in the case   1, m  5,   3 and   2 of
Example1.
Example 2.
We consider the following fractional Riccati differential equation
D y( x)  y( x)  y 2 ( x)  0,
0  x  1, 0    1
(5.4)
with initial condition
1
y (0)  .
2
(5.6)
The exact solution of this problem is
y ( x) 
e x
.
1  e x
By using the method that was elaborated in previous section, we have the approximations (4.5)(4.10).
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Using (4.11) and Tau method, the problems (5.4)-(5.5) are transformed to the following relation


K T Q(1  )  Q(1) H P 
Letting m  3 ,     1 and   0.8 , we obtain
Q
0.2 
1
 0.
4
(5.7)
 0.935031 0.0703778  0.00754058
  0.392111 0.727248
0.0446918 ,
 0.642244  0.437149
0.643899 
0.5
0.16667
0.0 

Q1   0.642857
0.0
0.0535714,
 1.33333
 0.437149
0.0 
H  h1 h2 h3 .
It is easy to see that
1


1  1

   1 0 0 P.
  
0 0 
6x  3

  4
4  4

28 x 2  28 x  6
Now, from (5.6) we conclude that K   63.4958 63.7014
 25.8788 . So,
y( x)  0.2351 x2  0.035180 x  0.37578.
 0.608934k0  0.881562k1  0.725841k2 
h1   0.214920k0  0.140508k1  1.41862k2 ,
 0.0110388k0  0.199772k1  0.0245617 k2 
 0.608934k0  0.881562k1  0.725841k2 
h2   0.167160k0  0.109284k1  1.103372k2 ,
 0.00858573k0  1.55379k1  0.190802k2 
 0.228350k0  0.330586k1  0.272190k2 
h3   0.0805950k0  0.526905k1  0.531984k2 ,
0.00413955k0  0.0749146k1  0.00919940k2 
T
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889
Figure 2. The approximate solution in the case   1, m  5,   3 and   2 of
Example2.
Example 3.
As a final example, we consider the following fractional Riccati differential equation
D y( x)  y 2 ( x)  x2 , 0  x  1, 0    1
(5.7)
with initial condition
y(0)  1.
(5.8)
The exact solution of this problem is
  x2   1 
 x2   3  
x J 3     J 3    
 2   4  4  2   4 
y ( x)   4 2
,
 x  1
 x2   3 
J 1     2 J 1   
2  4
2  4
4
4 
where J (t ) is the Bessel function of the first kind.
We suppose that m  3 ,     1 and  
13
x2  1  
15
1
. It is easy to see that
2
1


1 1 
13
4 x  2   


30 15 
15
15 x 2  2 x  3
1 1
P.
30 15 
In the same way as in the previous examples, by using (4.11), the problems (5.7)-(5.8) are
transformed to the following relation:
 ( 12 )
 13
K  Q  Q (1) H  2Q (1)   

 15
T
1
30
1
P.
15 
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A. Neamaty et al.
Now, using Tau method, we reduce the problem to solve the following system of algebraic
equation
 (1)
 13
K T  Q 2  Q(1) H  2Q(1)   

 15
1 1
.
30 15 
(5.9)
Now, from (5.9) we conclude
K  1.65702  2.45478 1.81873 .
T
So
y( x)  2.554778x 2  3.33234 x  2.79004.
In Table 1, the approximate solutions for test problems 1, 2 and 3 obtained by different values of
m,  ,  and  using the presented method.
Table 1. The approximate solutions for examples 1, 2 and 3
EX
m



y
2
0.5
1
1
1.000 x  6.1391 10 7
0.8
2
2
0.96249 x 2  0.80401 x  6.1391 10 7
0.9
1
3
0.21268x 3  0.65203 x 2  0.99962 x  7.2732  10 7
3
EX 1
4
 0.36169 x 4  0.73652 x 3  0.96249 x 2 
5
1.0
3
2
2
0.5
1
1
 0.029326 x  0.38318
0.8
2
2
0.23478 x 2  0.035191 x  0.38941
4
0.9
1
3
0.0031179 x 3  0.052465 x 2  0.032258 x  0.41260
5
1.0
3
2
2
0.5
1
1
0.0001230466 x  0.9931299
0.8
2
2
0.6529539 x 2  0.5156453 x  0.9761941
4
0.9
1
3
6.608623x 3  0.6529539 x 2  0.005250264 x  0.9931489
5
1.0
3
2
3
EX 2
3
EX 3
0.80401 x  7.1306  10 7
 .00027185 x 4  .00010234 x 3  .28132 x 2 
.0000032236 x  .52133
 14.36735 x 4  14.45792 x 3  .6528798 x 2
 1283269  10 5 x  .9931370
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891
5. Conclusion
In this paper, we have proposed a numerical method for solving Riccati differential equation of
fractional order. The shifted Jacobi polynomial integral operational matrix was developed to
solve this equation. The numerical results showed this method is powerful, new and interesting.
All of the numerical computations in this study have been done on a PC applying some programs
written in MAPLE.
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